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$P: (x,y,z) = (0,-7,0) + t_1(1,-2,0) + t_2(0,1,1)$ $t_1,t_2$ contained in R and the point $M(-1,0,-1)$

Find the point $N$ on the plane $P$ that is closest to point $M$. Also find the point $M'$ that is the reflection of the point $M$ with respect to the plane $P$.

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Idea: First find a normal vector $\vec{n}$ for the plane P.:

$$ \vec{n} = \pmatrix{1 \\ -2\\ 0 }\times \pmatrix{0\\1\\1} =\pmatrix{a \\ b \\ c}. $$

Then write down an equation for the plane in the form: $$ ax + b(y+7) + cz = 0. $$ Find parametric equations of the line that passes through $M$ with direction vector $\vec{n}$: $$\begin{align} x &= -1 + sa \\ y &= 0 + sb \quad\quad s\in\mathbb{R}\\ z &= -1 + sc. \end{align}$$ Find the intersection between the line and the plane. You can do this by putting the expressions for the line into the equation for the line above (i.e. you end up solving an equation for $s$). This intersection will be the point $N$.

Now find the distance $\lvert NM\rvert$. The "go that distance" on the other side of the plane using the parametric equations for the line. Or, find the intersection between the sphere of radius $\lvert NM\rvert$ with center $N$ and the line.

(Try to draw a picture to see why this works.)

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Thank you so much for your help! – Dragon Feb 14 '13 at 2:47
@Dragon: No problem. Glad to help. – Thomas Feb 14 '13 at 2:48

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